cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A213760 Antidiagonal sums of the convolution array A213783.

Original entry on oeis.org

1, 4, 12, 27, 52, 92, 148, 230, 335, 480, 656, 889, 1162, 1512, 1912, 2412, 2973, 3660, 4420, 5335, 6336, 7524, 8812, 10322, 11947, 13832, 15848, 18165, 20630, 23440, 26416, 29784, 33337, 37332, 41532, 46227, 51148, 56620, 62340, 68670
Offset: 1

Views

Author

Clark Kimberling, Jun 22 2012

Keywords

Links

Crossrefs

Cf. A213783, A213500.

Programs

  • Mathematica
    (See A213783.)
LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{1,4,12,27,52,92,148,230},40] (* Harvey P. Dale, Feb 13 2024 *)
  • PARI
    Vec(x*(1 + x - x^2)*(1 + x + 2*x^2) / ((1 - x)^5*(1 + x)^3) + O(x^60)) \\ Colin Barker, May 04 2017

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) -2*a(n-7) +a(n-8).
G.f.: f(x)/g(x), where f(x) = x*(1 + 2*x + 2*x^2 + x^3 -2*x^4) and g(x) = (1 + x)^3 *(1 - x)^5.
From Colin Barker, May 04 2017: (Start)
a(n) = (2*n^4 + 22*n^3 + 40*n^2 + 8*n) / 96 for n even.
a(n) = (2*n^4 + 22*n^3 + 34*n^2 + 26*n + 12) / 96 for n odd.
(End)

A213763 Principal diagonal of the convolution array A213762.

Original entry on oeis.org

1, 11, 43, 127, 331, 807, 1891, 4319, 9691, 21463, 47059, 102351, 221131, 475079, 1015747, 2162623, 4587451, 9699255, 20447155, 42991535, 90177451, 188743591, 394264483, 822083487, 1711275931, 3556769687, 7381974931
Offset: 1

Views

Author

Clark Kimberling, Jun 20 2012

Keywords

Comments

Create a triangle with first column T(n,1)=1+4*n for n=0,1,2... The remaining terms T(r,c)=T(r,c-1)+T(r-1,c-1). The sum of the terms in row(n)=a(n+1). - J. M. Bergot, Dec 18 2012

Links

Crossrefs

Cf. A213762, A213500.

Programs

  • Mathematica
    (See A213762.)
LinearRecurrence[{6,-13,12,-4},{1,11,43,127},30] (* Harvey P. Dale, Apr 13 2017 *)

Formula

a(n) = -1 + 2^n - 4*n + n*2^(n+1).
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).
G.f.: x*(1 + 5*x - 10*x^2)/(1 - 3*x + 2*x^2 )^2.

A213764 Antidiagonal sums of the convolution array A213762.

Original entry on oeis.org

1, 8, 31, 90, 225, 516, 1123, 2366, 4885, 9960, 20151, 40578, 81481, 163340, 327115, 654726, 1310013, 2620656, 5242015, 10484810, 20970481, 41941908, 83884851, 167770830, 335542885, 671087096, 1342175623, 2684352786
Offset: 1

Views

Author

Clark Kimberling, Jun 20 2012

Keywords

Links

Crossrefs

Cf. A213762, A213500.

Programs

  • Mathematica
    (See A213762.)
LinearRecurrence[{5,-9,7,-2},{1,8,31,90},30] (* Harvey P. Dale, Jan 27 2016 *)

Formula

a(n) = -10 + 5*2^(n+1) - 7*n - 2*n^2.
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4).
G.f.: f(x)/g(x), where f(x) = x*(1 + 3*x) and g(x) = (1 - 2*x)*(1 - x)^3.

A213766 Principal diagonal of the convolution array A213765.

Original entry on oeis.org

1, 5, 24, 83, 263, 776, 2201, 6077, 16488, 44211, 117615, 311232, 820641, 2158645, 5669096, 14872483, 38989367, 102165928, 267628569, 700924525, 1835493016, 4806144675, 12583939679, 32947361088, 86260987393, 225840387941
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Links

Crossrefs

Cf. A213765, A213500.

Programs

Formula

a(n) = 5*a(n-1) - 6*a(n-2) - 3*a(n-3) + 6*a(n-4) + a(n-5) - a(n-6).
G.f.: f(x)/g(x), where f(x) = x*(1 + 5*x - 10*x^2) and g(x) = (1 - 3*x + 2*x^2 )^2.

A213767 Antidiagonal sums of the convolution array A213765.

Original entry on oeis.org

1, 5, 17, 47, 114, 254, 533, 1071, 2083, 3951, 7348, 13452, 24313, 43481, 77077, 135615, 237094, 412234, 713325, 1229155, 2110151, 3610655, 6159912, 10481112, 17790769, 30132269, 50933273, 85936271, 144750618, 243438806
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Links

Crossrefs

Cf. A213765, A213500.

Programs

Formula

a(n) = 4*a(n-1)-4*a(n-2)-2*a(n-3)+4*a(n-4)-a(n-6).
G.f.: f(x)/g(x), where f(x) = x*(1 + x + x^2 + x^3) and g(x) = (1 - 2*x + x^3)^2.
a(n) = n*Fibonacci(n+5) - Lucas(n+6) + 2*(2*n+9). - Ehren Metcalfe, Jul 10 2019

A213769 Principal diagonal of the convolution array A213768.

Original entry on oeis.org

1, 8, 26, 63, 136, 272, 521, 968, 1762, 3159, 5600, 9840, 17169, 29784, 51418, 88399, 151432, 258592, 440345, 747960, 1267586, 2143783, 3618816, 6098208, 10260001, 17236712, 28918106, 48454623, 81093832, 135569264, 226404905
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Links

Crossrefs

Cf. A213768, A213500.

Programs

  • Mathematica
    (See A213768.)
LinearRecurrence[{4,-4,-2,4,0,-1},{1,8,26,63,136,272},40] (* Harvey P. Dale, Jan 08 2015 *)

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6).
G.f.: f(x)/g(x), where f(x) = x*(1 + 4*x - 2*x^2 - 7*x^3) and g(x) = (1 - 2*x + x^3 )^2.

A213770 Antidiagonal sums of the convolution array A213768.

Original entry on oeis.org

1, 7, 23, 58, 126, 250, 467, 837, 1457, 2484, 4172, 6932, 11429, 18739, 30603, 49838, 81002, 131470, 213175, 345425, 559461, 905832, 1466328, 2373288, 3840841, 6215455, 10057727, 16274722, 26334102, 42610594, 68946587, 111559197
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Links

Crossrefs

Cf. A213768, A213500.

Programs

  • Magma
    [2*Fibonacci(n+6)+Lucas(n+4)-n*(2*n+11)-23: n in [1..35]]; // Vincenzo Librandi, Jul 09 2019
  • Mathematica
    (See A213768.)

Formula

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: f(x)/g(x), where f(x) = x*(1 + 3*x) and g(x) = (1 - x - x^2)(1 - x)^3.
a(n) = 2*Fibonacci(n+6) + Lucas(n+4) - n*(2*n + 11) - 23. - Ehren Metcalfe, Jul 08 2019

A213775 Principal diagonal of the convolution array A213774.

Original entry on oeis.org

1, 11, 38, 97, 213, 432, 833, 1555, 2838, 5097, 9045, 15904, 27761, 48171, 83174, 143009, 244997, 418384, 712465, 1210195, 2050966, 3468681, 5855333, 9867072, 16600993, 27889547, 46790438, 78401185, 131212533, 219355632
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Links

Crossrefs

Cf. A213774, A213500.

Programs

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6).
G.f.: f(x)/g(x), where f(x) = x*(1 + 7*x - 2*x^2 - 9*x^3 - 5*x^4) and g(x) = (1 - 2*x + x^3 )^2.

A213776 Antidiagonal sums of the convolution array A213774.

Original entry on oeis.org

1, 8, 30, 81, 184, 376, 717, 1304, 2294, 3941, 6656, 11104, 18361, 30168, 49342, 80441, 130840, 212472, 344645, 558600, 904886, 1465293, 2372160, 3839616, 6214129, 10056296, 16273182, 26332449, 42608824, 68944696, 111557181
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Links

Crossrefs

Cf. A213774, A213500.

Programs

Formula

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: f(x)/g(x), where f(x) = x*(1 + 4*x + 3*x^2) and g(x) = (1 - x - x^2)*(1 - x)^3.
a(n) = 6*Fibonacci(n+6) - Lucas(n+5) - 2*n*(2*n+9) - 37. - Ehren Metcalfe, Jul 10 2019

A213777 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = F(h+1), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 3, 2, 7, 5, 3, 15, 12, 8, 5, 30, 25, 19, 13, 8, 58, 50, 40, 31, 21, 13, 109, 96, 80, 65, 50, 34, 21, 201, 180, 154, 130, 105, 81, 55, 34, 365, 331, 289, 250, 210, 170, 131, 89, 55, 655, 600, 532, 469, 404, 340, 275, 212, 144, 89, 1164, 1075, 965, 863
Offset: 1

Views

Author

Clark Kimberling, Jun 21 2012

Keywords

Comments

Principal diagonal: A001870
Antidiagonal sums: A152881
row 1, (1,1,2,3,5,8,...)**(1,2,3,5,8,13,...): A023610(k-1)
row 2, (1,1,2,3,5,8,...)**(2,3,5,8,13,21,...): A067331(k-1)
row 3, (1,1,2,3,5,8,...)**(3,5,8,13,21,34,...)
For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....3....7....15....30....58
2....5....12...25....50....96
3....8....19...40....80....154
5....13...31...65....130...250
8....21...50...105...210...404
13...34...81...170...340...654

Links

Crossrefs

Cf. A213500.

Programs

  • Mathematica
    b[n_] := Fibonacci[n]; c[n_] := Fibonacci[n + 1];
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213777 *)
Table[t[n, n], {n, 1, 40}] (* A001870 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A152881 *)

Formula

T(n,k) = 2*T(n,k-1) + T(n,k-2) - 2*T(n,k-3) - T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = F(n-1) + F(n-2)*x and g(x) = (1 - x - x^2)^2.
T(n,k) = (k*Lucas(n+k+1) + Lucas(n)*Fibonacci(k))/5. - Ehren Metcalfe, Jul 10 2019
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